October, 1997

The article grew out of the author's playing around with randomly chosen line segments in a convex domain and the probability that they intersect. It turned out that this probability is connected to an old problem posed by Sylvester: What is the probability that four independently chosen points in a convex bounded domain span a quadrilateral?

The author derives a surprising relationship between areas of particular subsets of an arbitrary bounded convex domain by interpreting Sylvester's probability in two different ways. He also considers a three-dimensional analogue of his original question: Assume a triangle and a line segment are chosen at random in the 3-dimensional unit ball. What is the probability that they intersect?

The prime number theorem, now celebrating its 100th birthday, is one of the most famous results in number theory, but has always retained an aura of mystery because there were no really easy proofs: the original proofs given by Hadamard and by de la Vallee Poussin in 1896, and many subsequent variants, involved complicated and required proving non-trivial lower bounds on the size of the Riemann zeta-function on the line Re(s)=1; other analytic proofs avoided such estimates but needed instead subtle Tauberian theorems like the Wiener-Ikehara theorem; and the "elementary" proof given by Selberg and Erdos in 1949 was far more complicated than the analytic proofs and, despite various subsequent simplifications, has remained so to this day. In 1980 a very simple replacement of the needed Tauberian argument was discovered by D.J. Newman. The resulting proof of the prime number theorem is short, beautiful, and understandable without any knowledge of number theory or complex function theory beyond Cauchy's theorem; it should be known to every mathematician. The present article is intended as a step towards this goal.

An Exploratory Approach to Kaplansky's Lemma Leads to a Generalized Resultant

A generalization of the classical notion of resultant from two to several polynomials is given. An immediate consequence is Kaplansky's Lemma on linear operators for arbitrary fields, at least in the finite-dimensional case.

Clearly any subspace of the real line with the Euclidean metric is a metric space in which all triangles are degenerate. Can you find the other metric spaces in which all triangles are degenerate? Is the usual topology on the plane generated by any metric in which all triangles are degenerate? Is the usual topology on the real line generated by any metric which has no degenerate triangles with more than two distinct vertices? These questions are answered in our article.

Existence of partitions of unity for metric spaces is usually proved using the (equivalent) concept of paracompactness. Although the standard proof of paracompactness of metric spaces is the one given by M.E. Rudin, in his 1965 PhD thesis, Michael Mather showed that it is easier, for metric spaces, to show directly the existence of locally finite partitions of unity for arbitrary covers. We adapt Mather's argument to prove existence of a partition of unity subordinated to a countable open cover of a metric space. An advantage of the method is that the same proof can be used in the smooth category.

Curve-fitting, cubic splines, regression lines, symbolic derivatives -- it's as if the computer was made for doing calculus. Anywhere calculus is found, a computer is not far away -- except, that is, in a calculus course. Could it be that our calculus curriculum is out of step with the science and mathematics of this century? If so, should not that curriculum be changed? And if it is to be changed, which topics should go and which should stay? This essay addresses such questions as it investigates what should constitute a modern calculus curriculum.

Although self-styled reformers tend to agree on a few common broad goals and tenets, such as the primacy of conceptual understanding, there is no single calculus reform "party line." On the contrary, different reformers make significantly different choices --- indeed, reformed approaches offer more diversity of choice than do traditional texts. Calculus reform is not a single alternative to a standard diet; reform offers a diverse menu of different but carefully-considered choices. The greatest lasting value of calculus reform may well be in highlighting and forcing important choices among competing goods --- choices that have always been present but too little acknowledged. In the end, choosing everything means choosing nothing.

This paper addresses a new level of K-16 Calculus reform by first distinguishing between that peculiar American web of habits, expectations, and structures termed "Calculus the Institution" and that splendid historical achievement, "Calculus the System of Knowledge and Technique." Calculus Reform has been about remodeling the former while leaving the larger K-12 educational structures in place, which in turn continues to deny most students (90%) access to the key aspects of the latter. We argue that a deeper reform is in order, one that deeply rethinks the subject matter of calculus in terms of learning and thinking, that makes intelligent and innovative use of potent technologies, and that integrates the mathematics of change and variation with other important mathematics across grades K-12. The author illustrates efforts in these directions through the NSF funded SimCalc Project (http://www.simcalc.umassd.edu), which is directed towards democratizing access to the mathematics of change and variation beginning in the middle grades and even earlier.

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